Theorem fusgrvtxfi 29957

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 29956	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 503	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6566  Fincfn 9011  Vtxcvtx 29631  USGraphcusgr 29787  FinUSGraphcfusgr 29954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-fusgr 29955

This theorem is used by:  fusgrfupgrfs  29969  nbfusgrlevtxm1  30015  nbfusgrlevtxm2  30016  nbusgrvtxm1  30017  uvtxnm1nbgr  30042  cusgrm1rusgr  30220  wlksnfi  30553  fusgrhashclwwlkn  30727  clwwlkndivn  30728  fusgreghash2wsp  30996  numclwwlk3lem2  31042  numclwwlk4  31044